Small data scattering of the inhomogeneous cubic-quintic NLS in 2 dimensions

TL;DR

This paper proves small data scattering for the 2D inhomogeneous cubic-quintic nonlinear Schrödinger equation using angular regularity and Strichartz estimates, and discusses conditions for non-scattering.

Contribution

It introduces a new approach to small data scattering in 2D ICQNLS using angular regularity and averaged Strichartz estimates, extending previous results.

Findings

Proved small data scattering in $H_ heta^{1,1}$ space.

Established decay properties of angularly regular functions.

Provided conditions for non-existence of scattering.

Abstract

The aim of this paper is to show the small data scattering for 2D ICQNLS: $$iu_t=-\Delta u + K_1(x)|u|^2u+K_2(x)|u|^4u.$$ Under the assumption that $\left| \partial^j K_l \right| \lesssim |x|^{b_l -j}$ for $j=0, 1, 2, l=1, 2$ and $0 \le b_l \le l - \frac23$, we prove the small data scattering in an angularly regular Sobolev space $H_\theta^{1,1}$. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space $H_\theta^{1, 1} \subset H^1$ with angular regularity such that $\|\partial_\theta f\|_{H^1} < \infty$, and also use the recently developed angularly averaged Strichartz estimates \cite{stri2, cholee, ghn}. In addition, we suggest a sufficient condition for non-existence of scattering.